4 papers
Variationally correct loss functions for nonlinear parametric problems and their neural approximations
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We construct computable a posteriori error estimators for parametric families of nonlinear elliptic problems by measuring the residual of a variational formulation in the dual norm…
Global error estimators for parametric monotone nonlinearities and neural approximations
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We construct computable error estimators, which double as loss functions for neural networks, for a class of parametric nonlinear partial differential equations with a monotonicity…
Fortin operators for DPG advection discretizations
Pablo Cortés Castillo, Jay Gopalakrishnan
We construct Fortin operators for discontinuous Petrov-Galerkin (DPG) discretizations of the advection equation with a piecewise constant divergence-free advection vector , on s…
DPG loss functions for learning parameter-to-solution maps by neural networks
Pablo Cortés Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of p…